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Answer :
To find the value of [tex]\(\log_5 625\)[/tex], we need to determine what power we must raise 5 to in order to get 625. This can be expressed by the equation:
[tex]\[ 5^x = 625 \][/tex]
We need to solve for [tex]\(x\)[/tex]. To do this, let's consider the prime factorization of 625:
1. Start dividing 625 by 5, since 5 is the base of our logarithm:
- [tex]\(625 \div 5 = 125\)[/tex]
2. Continue dividing by 5:
- [tex]\(125 \div 5 = 25\)[/tex]
3. Divide by 5 again:
- [tex]\(25 \div 5 = 5\)[/tex]
4. Finally, divide by 5 once more:
- [tex]\(5 \div 5 = 1\)[/tex]
We divided a total of four times by 5 to simplify 625 to 1. This shows that:
[tex]\[ 625 = 5 \times 5 \times 5 \times 5 = 5^4 \][/tex]
Thus, [tex]\(5^4 = 625\)[/tex], meaning the exponent [tex]\(x\)[/tex] is equal to 4. Therefore, the value of [tex]\(\log_5 625\)[/tex] is:
[tex]\[ \log_5 625 = 4 \][/tex]
[tex]\[ 5^x = 625 \][/tex]
We need to solve for [tex]\(x\)[/tex]. To do this, let's consider the prime factorization of 625:
1. Start dividing 625 by 5, since 5 is the base of our logarithm:
- [tex]\(625 \div 5 = 125\)[/tex]
2. Continue dividing by 5:
- [tex]\(125 \div 5 = 25\)[/tex]
3. Divide by 5 again:
- [tex]\(25 \div 5 = 5\)[/tex]
4. Finally, divide by 5 once more:
- [tex]\(5 \div 5 = 1\)[/tex]
We divided a total of four times by 5 to simplify 625 to 1. This shows that:
[tex]\[ 625 = 5 \times 5 \times 5 \times 5 = 5^4 \][/tex]
Thus, [tex]\(5^4 = 625\)[/tex], meaning the exponent [tex]\(x\)[/tex] is equal to 4. Therefore, the value of [tex]\(\log_5 625\)[/tex] is:
[tex]\[ \log_5 625 = 4 \][/tex]
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